The aim of this note is to study the magnitude of the constants in the equivalence between the first and secondorder Ditzian-Totik moduli of smoothness and related K-functionals. Applications to some classic approximationoperators are given

Tachev, Gancho T.

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Aurel Vlaicu University of Arad Editing House: Journals

Theory and Applications of Mathematics & Computer Science 2 (2) (2012) 48–54Refined Estimates for the Equivalence Between Ditzian-TotikModuli of Smoothness and K-FunctionalsGancho T. Tacheva,∗aUniversity of Architecture, Civil Engineering and Geodesy, Department of Mathematics, BG-1046 Sofia, BulgariaAbstractThe aim of this note is to study the magnitude of the constants in the equivalence between the first and secondorder Ditzian-Totik moduli of smoothness and related K-functionals. Applications to some classic approximationoperators are given.Keywords: Ditzian-Totik moduli of smoothness, K-functional, smooth functions, constants in the equivalence.2010 MSC: 41A15, 41A25, 41A28, 41A10.1. Introduction and main resultsDitizian-Totik moduli of smoothness have become a standard tool in approximation theory.This is true in particular for second order moduli which play a crucial role in approximation bypositive linear operators. For their properties and many applications see (Z. Ditzian, 1987). Forf ∈ C[0, 1] the second order Ditzian-Totik modulus is defined byωϕ2( f , h) = sup{|∆2ρϕ(x) f (x)|, x ± ρϕ(x) ∈ [0, 1], 0 < ρ ≤ h.} (1.1)Here ϕ(x) =√x(1 − x), and ∆2η f (y) = f (y − η) − 2 f (y) + f (y + η) if η > 0 and y ± η ∈ [0, 1] andas 0 otherwise. In the sequel we will use the following notation:ACloc[0, 1] := {h : h is absolutely continuous in [a, b] for every 0 < a < b < 1};Wϕ2,∞[0, 1] := {g : g′ ∈ ACloc[0, 1] and ‖ϕ2g′′‖∞ < ∞}.The related K-functional Kϕ2 ( f , h2) is given byKϕ2 ( f , h2) := infg{‖ f − g‖∞ + h2‖ϕ2g′′‖∞}. (1.2)∗Corresponding authorEmail address: gtt−fte@uacg.acad.bg (Gancho T. Tachev)Gancho T. Tachev / Theory and Applications of Mathematics & Computer Science 2 (2) (2012) 48–54 49Here the infimum is taken over all f ∈ Wϕ2,∞[0, 1]. The definitions of the second order Ditzian-Totikmodulus of smoothness and related K-functional can be generalized in a natural way for all r ≥ 1.The equivalence between these two constructive characteristics is well-known (see Theorem 6.2in (DeVore & Lorentz, 1993)). We cite it here asTheorem A There are constants c1, c2 > 0, which depend only on r, such that for all f ∈ C[0, 1]c1ωϕr ( f , h) ≤ Kϕr ( f , hr) ≤ c2ωϕr ( f , h), 0 < h ≤ (2r)−1. (1.3)In many problems in approximation theory it is a difficult task to prove direct or inverseestimates directly in terms of the Ditzian-Totik moduli of smoothness. Instead of this, the K-functionals have become a powerful tool to establish such statements. However, the main dis-advantage of the latter, is the fact, that practically it is impossible to calculate the value of theK-functional for a given function f . Therefore, the usual way is first to prove direct or inverseestimates in terms of the K-functional, and after this using Theorem A to reformulate the resultsin terms of the moduli of smoothness. This explains how important is to have a good informationabout the magnitude of the constants c1, c2. To the best of our knowledge the problem to findthe best possible values of c2 and c1 (in the sense c2-minimal and c1-maximal in (1.3)) is still notsolved. Hardly anything appears to be known about the explicit description of the size of c1 andc2 for r = 1, 2 and about their asymptotic dependence on r for r > 2. The first attempt in thisdirection is Theorem 3.5 in (Gonska & Tachev, 2003) which we cite here asTheorem B For m ≥ 2, h ∈[ √2md(m) ,√2(m−1)d(m−1)]the following inequalities hold for any f ∈C[0, 1]:116ωϕ2( f , h) ≤ Kϕ2 ( f , h2) ≤ c2(m)ωϕ2( f , h),wherec2(m) := 1 +( mm − 1)2 48d2(m − 1),and the sequence d(m) is defined asd(m) =√m4 + m2 + 1 − 1√m4 + m2 + 1 + m2, d(m)→12, m→ ∞.It is clear that limm→∞c2(m) = 193. If we restrict our attention to values h ≤ 1, as a corollary fromTheorem B we get116ωϕ2( f , h) ≤ Kϕ2 ( f , h2) ≤ 404 · ωϕ2( f , h). (1.4)The difficulties in the proof of Theorem B are connected with the construction of an appropriateauxiliary function g in the definition of Kϕ2 . Actually we apply a ”smoothing” technique to the lin-ear interpolant on certain places near the points of interpolation to obtain an appropriate quadraticC1-spline based upon the knot sequence. This method was developed in (Gonska & Kovacheva,1994; Gonska & Tachev, 2003; H. Gonska, 2002) and further refined in (Gavrea, 2002). In this50 Gancho T. Tachev / Theory and Applications of Mathematics & Computer Science 2 (2) (2012) 48–54note we essentially improve the value of the constant 404 in (1.4). Our main result states the fol-lowing:Theorem 1 The following inequalities hold for any f ∈ C[0, 1], h ∈ (0, 1]:116ωϕ2( f , h) ≤ Kϕ2 ( f , h2) ≤ (5 + 2√2) · ωϕ2( f , h). (1.5)In Section 2 we give the proof of Theorem 1. In Section 3 we apply Theorem 1 to obtainquantitative estimates in terms of second order Ditzian-Totik modulus of smoothness for approx-imation by genuine Bernstein-Durrmeyer operator, considered in (P. E. Parvanov, 1994) and alsofor pointwise estimates, established in (Felten, 1998). In the last section we consider the caser = 1, which is closely related to piecewise linear interpolation at specific knot sequence.2. Proof of Theorem 1To obtain as small as possible value of the constant c2 in (1.3) we need an appropriate auxiliaryfunction g in the definition of the K-functional. We use the construction, developed by Gavreain (Gavrea, 2002) and based on the ideas from (Gonska & Kovacheva, 1994; Gonska & Tachev,2003). Let m be fixed natural number, m ≥ 1. The patition ∆m of the interval [0, 1] is given by∆m : 0 = x0 < x1 < · · · < x2m+2 = 1,wherexk = sin2kπ4(m + 1), k = 0, 1, . . . , 2m + 2. (2.1)We denote by S m( f ) a piecewise linear interpolant with interpolation knots-the points xk, k =0, 1, . . . , 2m + 2. Each point (xk, S m( f , xk)), k = 1, 2, . . . , 2m + 1 we associate with two other points(ak, S m( f , ak)), (bk, S m( f , bk)) such thata1 =x12, b1 − x1 = x1 − a1,andak =xk + xk−12, bk − xk = xk − ak, k = 1, 2, . . . , 2m + 1.The function g is defined as follows:For x ∈ [0, a1] ∪ [b2m+1, 1] we set g(x) = S m( f , x).For x ∈ [ak, bk], k = 1, . . . , 2m + 1, g(x) is the 2nd degree Bernstein polynomial over theinterval [ak, bk], determined by the ordinates S m( f , ak), f (xk), S m( f , bk).For x ∈ [bk, ak+1], k = 1, 2, . . . , 2m we set g(x) = S m( f , x). Thus g(x) is uniquely determinedby the interpolation conditions and is C1-continuous. For this function the following two crucialestimates are proved in Theorem 6 in (Gavrea, 2002):‖ f − g‖∞ ≤ ωϕ2(f , sinπ2(m + 1)), (2.2)Gancho T. Tachev / Theory and Applications of Mathematics & Computer Science 2 (2) (2012) 48–54 51‖ϕ2g′′‖∞ ≤1sin2 π4(m+1)· ωϕ2(f , sinπ2(m + 1)). (2.3)For any positive number h ∈ (0, 1] there exists a natural number m ≥ 1, such thath ∈[sinπ2(m + 1), sinπ2m].Hence (2.2) and (2.3) imply‖ f − g‖∞ ≤ ωϕ2( f , h), (2.4)h2‖ϕ2g′′‖∞ ≤sin2 π2msin2 π4(m+1)· ωϕ2( f , h). (2.5)It is easy to verify that the sequencec(m) :=sin2 π2msin2 π4(m+1)is monotone decreasing, i.e. c(m) ≤ c(1) = 4 + 2√2. Consequently the right-hand side of (1.5)is proved. Lastly we point out that the constant 116 in (1.5) could be derived from Theorem 6.1 in(DeVore & Lorentz, 1993). Thus the proof of Theorem 1 is completed.3. Applications1. The genuine Bernstein-Durrmeyer operator. As first application of Theorem 1 let us con-sider the so-called genuine Bernstein-Durrmeyer operator, introduced by Goodman and Sharma in(Goodman & Sharma, 1991) and given byUn( f , x) = f (0)pn,0(x) + f (1)pn,n(x) + (n − 1)n−1∑k=1pn,k(x)∫ 10pn−2,k−1(t) f (t)dt,where pn,k(x) =(nk)xk(1− x)n−k, k = 0, . . . , n, are the fundamental Bernstein polynomials. Parvanovand Popov proved in (P. E. Parvanov, 1994) in an elementary and very elegant manner a direct anda strong converse inequality of type A, thus completely characterizing the approximation speed ofthe operators. The main result in (P. E. Parvanov, 1994) states the following:For any f ∈ C[0, 1] we have12‖Un f − f ‖∞ ≤ Kϕ2 ( f ,12n) ≤ (4 +√2)‖Un f − f ‖∞. (3.1)As a corollary from Theorem 1 and (3.1) we obtain12(5 + 2√2)‖Un f − f ‖∞ ≤ ωϕ2( f ,1√2n) ≤ 16(4 +√2)‖Un f − f ‖∞. (3.2)52 Gancho T. Tachev / Theory and Applications of Mathematics & Computer Science 2 (2) (2012) 48–542. The Bernstein operator The classical Bernstein operator Bn( f , x) for a given functionf ∈ C[0, 1] is defined byBn( f , x) =n∑k=0f (kn)pn,k(x).Let Φ : [0, 1] → R, Φ , 0 be a function such that Φ2 is concave. Then the pointwise approxima-tion|Bn( f , x) − f (x)| ≤ 2Kϕ2(f , n−1ϕ2(x)Φ2(x)), x ∈ [0, 1], (3.3)holds true for all f ∈ C[0, 1], n ∈ N. This result was proved by Felten in (Felten, 1998). As astraightforward corollary from Theorem 1 we get|Bn( f , x) − f (x)| ≤ 2(5 + 2√2)ωϕ2(f , n−12ϕ(x)Φ(x)), x ∈ [0, 1]. (3.4)4. The case r = 1In this section we consider the interval [−1, 1] instead of [0, 1]. After a linear transformationit is clear that each estimate in one of these two cases can be obtained from the other. The weightfunction over [−1, 1] is now ϕ(x) =√1 − x2. Let ∆n : −1 = x0 < x1 < · · · < xn = 1 be a partitionof the interval [−1, 1] such that the inequalitiesc3(xk+1 − xk) ≤ϕ(x)n≤ c4(xk+1 − xk) (4.1)are satisfied for k = 1, 2, . . . , n − 2, x ∈ [xk, xk+1], and alsoc3(x + 1) ≤ϕ(x)n≤ c4(x + 1), x ∈ [x0, x1],c3(1 − x) ≤ϕ(x)n≤ c4(1 − x), x ∈ [xn−1, xn],where ci, i = 3, 4 are absolute positive constants independent of n. The function g in the definitionof Kϕ1 we define as the linear interpolant of f ∈ C[−1, 1] with knots {xk}. For x ∈ [xk, xk+1], k =1, . . . , n − 2, from the properties of linear interpolation it follows that|g(x)− f (x)| ≤ ω1( f , xk+1−xk) = sup{| f (x+h2)− f (x−h2)|, x, x±h2∈ [xk, xk+1]} ≤ ωϕ1( f ,1c3n). (4.2)Let x ∈ [−1, x1]. The case x ∈ [xn−1, 1] is analogous. Obviously|g(x) − f (x)| ≤ sup{| f (x +h2) − f (x −h2)|, x, x ±h2∈ [−1, x1]}.The inequality x − h2 ≥ −1 yields h ≤ 2(x + 1) ≤2c3ϕ(x)n , which follows from (4.1). To summarizewe proved‖ f − g‖∞ ≤ ωϕ1( f ,2c3n). (4.3)Gancho T. Tachev / Theory and Applications of Mathematics & Computer Science 2 (2) (2012) 48–54 53Next we evaluate the second term in the definition of the K-functional.For x ∈ [xk, xk+1], k = 1, 2, . . . , n − 2, it is easy to verify that1n|ϕ(x)g′(x)| =ϕ(x)n(xk+1 − xk)| f (xk) − f (xk+1)|.Using (4.1) and (4.2) we get1n‖ϕg′‖L∞[x1,xn−1] ≤ c4ωϕ1( f ,1c3n).It remains to consider x ∈ [−1, x1]. In this case we observe thatϕ(x)n≤ c4(x + 1).Therefore for x ∈ [−1, x1] we have1n|ϕ(x)g′(x)| ≤ c4ωϕ1( f ,2c3n).Finally we arrive at1n‖ϕg′‖ ≤ c4ωϕ1( f ,2c3n). (4.4)For every 0 < t < 1 there exists n ≥ 2 such that2c3n< t <2c3(n − 1).Combining (4.3) and (4.4) we getKϕ1 ( f , t) ≤[1 +2c4c3( nn − 1)]ωϕ1( f , t). (4.5)It is clear that the condition number c4c3 of our system of knots determines the value of the constantin front of the modulus. By the previous considerations we have shown the validity ofTheorem 2. For f ∈ C[−1, 1], n ≥ 2, t ≤ 12 we have18ωϕ1( f , t) ≤ Kϕ1 ( f , t) ≤ c2(n)ωϕ1( f , t), (4.6)wherec2(n) =[1 +2c4c3( nn − 1)].Remark 1. The constant 18 in the left side of (4.6) follows easily if we verify the computationsmade in Theorem 6.1 in Chapter 6 in (DeVore & Lorentz, 1993).54 Gancho T. Tachev / Theory and Applications of Mathematics & Computer Science 2 (2) (2012) 48–54Remark 2. If we strictly follow the construction in the proof of Theorem 2, it is possible to im-prove the value of c2(n), i.e. to obtain the value of the latter as small as possible. In order to dothis, we would find an optimal set of knots, satisfying (4.1) with a condition number as small aspossible. In this case we formulate the followingOpen problem. Find the optimal set {xk} satisfying (4.1) in the sense that the condition number c4c3is minimal.Here we give two examples of knots.Example A. In this example we choose {xk} to be the well-known zeros of the Chebyshevpolynomial of the first kindxk = cos θk, θk :=(2k − 1)π2n, k = 1, . . . , n, xn+1 := −1, x0 := 1.Following (7.7-7.8) in Chapter 8 in [1] we get c3 = 13π , c4 = 3π. The condition number is 9π2.Example B. Here we choose the extremal points of the Chebyshev polynomial of the first kindxk = cos(kπn), k = 0, . . . , n.This is the same set of interpolation knots , considered in Section 2 for the interval [0, 1]. In thiscase we compute c3 = 12π , c4 = 2π. The condition number is 4π2-better as in Example A.ReferencesDeVore, R. A. and G. G. Lorentz (1993). Constructive Approximation. Springer, Berlin.Felten, M. (1998). Direct and inverse estimates for Bernstein polynomials. Constructive Approximation 14, 459–468.Gavrea, I. (2002). Estimates for positive linear operators in trems of the second order Ditzian-Totik modulus ofsmoothness. Rend. Circ. Mat. Palermo (2) Suppl. (68), 439–454.Gonska, H. and R. Kovacheva (1994). The second order modulus revisited: remarks, applications, problems. In:Proceedings of the Confer. Sem. Mat. Univ. Bari. pp. 1–32.Gonska, Heinz H. and Gancho T. Tachev (2003). The second Ditzian-Totik modulus revisited: refined estimates forpositive linear operators.. Rev. Anal. Numér. Théor. Approx. 32(1), 39–61.Goodman, T. N. T. and A. Sharma (1991). A Bernstein-type operator on the Simplex. Math. Balkanica 5(2), 129–145.H. Gonska, G. Tachev (2002). On the constants inωϕ2- inequalities. Rend. Circ. Mat. Palermo (2) Suppl. (68), 467–477.P. E. Parvanov, B. D. Popov (1994). The limit case of Bernstein’s operators with Jacobi-weights. Math. Balkanica8(2-3), 165–177.Z. Ditzian, V. Totik (1987). Moduli of Smoothness. Springer, New York.

Refined Estimates for the Equivalence Between Ditzian-Totik Moduli of Smoothness and K-Functionals

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Refined Estimates for the Equivalence Between Ditzian-Totik Moduli of Smoothness and K-Functionals

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